It is known from either Morse theory or Bialynicki-Birula decomposition that the fixed points of a ${\mathbb{C}}^*$ action on a smooth algebraic variety over $\mathbb{C}$ determine a cell decomposition of this variety (cells correspond to fixed points by sending each point to a fixed point by the ${\mathbb{C}}^*$ action). Such cell decomposition of the Hilbert scheme of points on the plane $({\mathbb{C}}^2)^{[n]}$ is described in several sources, notably the original papers by G. Ellingsrud and S.A. Stromme and Chapter 5 of the book about Hilbert schemes of points on surfaces by H.Nakajima.

It follows that the fixed points of the torus action (and hence the cells) on $({\mathbb{C}}^2)^{[n]}$ are indexed by partitions of $n$. However, it is hard for me to infer from the above references what is the combinatorial order of these cells (i.e. which cells lie on the boundary of which other cells) -- this should induce a certain order on the partitions of $n$. Is there a reference which would state what this order is explicitly?